In a large saucepan, over medium heat, melt ½ cup of the butter. Add the oats and cook until golden, about 5 minutes. Spread on a baking sheet to cool.
Using an electric mixer on high, beat the remaining butter and the sugar until light and fluffy. Reduce speed to medium and add the peanut butter and vanilla extract. Reduce to low and slowly add the flour. Stir in the oats. Cover and refrigerate for 30 minutes.
Heat oven to 375° F. Form 16 to 20 large cookies, using about ¼ cup of dough for each. Place on parchment-or foil-lined baking sheets, 2 inches apart. Flatten slightly using the tines of a fork.
Bake until golden but not quite set, about 20 minutes. Let cool on the sheets for 10 minutes. Transfer to a rack to finish cooling.
Sandwich about 1/3 cup of the ice cream between 2 cookies. Serve immediately, or wrap in plastic and freeze.